Evaluating inverse trigonometric functions Without using a cal...

Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$an^{-1}\left( an\left(\frac{3\pi}{4} ight) ight)$

Accepted Answer

Welcome back, everyone. In this problem, we want to find the value of the inverse tangent of the tangent of 7 eighths of pi in terms of pi. For our answer choices, a is 7 8ths of Pi. B is negative 7 8ths of Pi. C is an 8th of Pi. And, d is a negative 8th of Pi. Now, what are we trying to do here? Well, we're trying to find the inverse tangent that's somehow related to 7 eighths of Pi. But the problem is 7 eighths of pi is not in the range of the inverse tangent. Let's draw that to make sense of what we have here. So if we think about our unit circle and have our X and Y axes and we have our angle 7 8th of pi. It's probably somewhere right here in our second quadrant. Then the range for the inverse tangent falls between a half of Pi and a negative half of Pi. So when we visualize it, you can see as well that 7 eighths of Pi does not fall in that range. So we need to find a reference angle for 7 eighths of Pi that is in the range for a half of Pi to a negative half of Pi. How are we going to do that? Well, recall that to find the reference angle for the tangent. So the tangent of X, we can add or subtract pi. So the tangent of X equals the tangent of X plus or minus pi. In this case, it would be better to subtract pi because that would likely bring us into our region that we need our angle. So what this tells us then is that the tangent of 7 eighths of pi would be equal to the tangent of 7 eighths of pi minus pi. Now, if we think about it, 7 eighths of pi minus pi, we can work that as a fraction. So we write both as a common denominator of 8, and that would be 7 Pi minus 8 Pi and 7 Pi minus 8 Pi is negative Pi. So, therefore, that tells us that the tangent of 7 eighths of Pi is the same thing as the tangent of a negative eighth of pi. And if we visualize that, it makes sense because a negative eighth of pi would probably be somewhere in our 4th quadrant and that would have been 180 degrees away or pi radians away rather. Now, since we know that, that means we can rewrite our problem of the inverse tangent of the tangent of 7 eighths of Pi as the inverse tangent of the tangent of negative a negative eighth of Pi. Now where do we go from here? Well, let's think about what we're doing. We're putting our function into its own inverse. And when we put our function into its own inverse, recall that that would give us the result or it would give us the input of our function, the value we started with. And that makes sense because if you think about it, if you take a value for a trig function, let's say in this case, it would be an angle and apply the trig function, then it's going to give you a ratio for that function. So if we want to find the angle that we started with, then we apply the inverse function to the ratio to get the angle. And that's the same idea here. So using that idea of putting our function into its own inverse, it stands to reason then. That the tangent or the inverse tangent of the tangent of a negative eighth of Pi would give us the angle we'd be starting with. In this case, a negative eighth of Pi. Therefore, when we look back on our answer choices, d would be the correct answer. A negative 8th of Pi. Thanks a lot for watching everyone. I hope this video helped.

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
$\tan^{-1}\left(\tan\left(\frac{3\pi}{4}\right)\right)$

Key Concepts

Inverse Trigonometric Functions

Periodic Functions

Angle Reduction

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